Question

misha found that the equation -|2x - 10| - 1 = 2 had two possible solutions: x = 3.5 and x = -6.5. which explains whether or not her solutions are correct? she is correct, because both solutions satisfy the equation. she is not correct, because she made a sign error. she is not correct, because there are no solutions. she is not correct, because there is only one solution: x = 3.5.

Explanation:

Step1: Isolate the absolute - value term

First, rewrite the given equation \(-|2x - 10|-1 = 2\). Add 1 to both sides of the equation:
\(-|2x - 10|=2 + 1\), so \(-|2x - 10|=3\). Then multiply both sides by - 1 to get \(|2x - 10|=-3\).

Step2: Analyze the property of absolute - value

The absolute - value of any real number \(a\), denoted as \(|a|\), is defined as \(|a|=

$$\begin{cases}a, & a\geq0\\-a, & a < 0\end{cases}$$

\), and \(|a|\geq0\) for all real numbers \(a\).
Since the left - hand side \(|2x - 10|\) is non - negative for all real \(x\) and the right - hand side is \(-3<0\), there are no real numbers \(x\) that satisfy the equation \(|2x - 10|=-3\).

Answer:

She is not correct, because there are no solutions.